Now the DNS records for domain1 point to the server's IP, this includes of course the MX record. The DNS record for domain2 only has the MX record pointing at the server's IP. However, using the settings described by dbmail, upon receiving an email for the above aliases/usernames Postfix Rejects the email saying "relay access denied".

I managed to sort this out by creating a new database table for allowed domains for the mydestination field. Now however, though it receives the mail - it bounces it and forwards it to the root account.